Properties

Label 1-4033-4033.3582-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.961 + 0.276i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.0581 − 0.998i)3-s + (0.766 − 0.642i)4-s + (0.993 + 0.116i)5-s + (0.396 + 0.918i)6-s + (0.597 − 0.802i)7-s + (−0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.973 + 0.230i)10-s + (−0.973 − 0.230i)11-s + (−0.686 − 0.727i)12-s + (0.396 + 0.918i)13-s + (−0.286 + 0.957i)14-s + (0.0581 − 0.998i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.0581 − 0.998i)3-s + (0.766 − 0.642i)4-s + (0.993 + 0.116i)5-s + (0.396 + 0.918i)6-s + (0.597 − 0.802i)7-s + (−0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.973 + 0.230i)10-s + (−0.973 − 0.230i)11-s + (−0.686 − 0.727i)12-s + (0.396 + 0.918i)13-s + (−0.286 + 0.957i)14-s + (0.0581 − 0.998i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.961 + 0.276i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3582, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.961 + 0.276i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.094154193 + 0.1541829005i\)
\(L(\frac12)\) \(\approx\) \(1.094154193 + 0.1541829005i\)
\(L(1)\) \(\approx\) \(0.7929573712 - 0.1020218959i\)
\(L(1)\) \(\approx\) \(0.7929573712 - 0.1020218959i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (-0.0581 - 0.998i)T \)
5 \( 1 + (0.993 + 0.116i)T \)
7 \( 1 + (0.597 - 0.802i)T \)
11 \( 1 + (-0.973 - 0.230i)T \)
13 \( 1 + (0.396 + 0.918i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.973 + 0.230i)T \)
31 \( 1 + (0.835 + 0.549i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (0.993 - 0.116i)T \)
53 \( 1 + (0.686 + 0.727i)T \)
59 \( 1 + (-0.0581 + 0.998i)T \)
61 \( 1 + (0.686 + 0.727i)T \)
67 \( 1 + (-0.973 - 0.230i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (-0.686 + 0.727i)T \)
79 \( 1 + (-0.286 - 0.957i)T \)
83 \( 1 + (0.396 + 0.918i)T \)
89 \( 1 + (0.0581 + 0.998i)T \)
97 \( 1 + (-0.893 - 0.448i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.23841545531314443582527833789, −17.73950962057034433729391725791, −17.363157685592016390389502208598, −16.3897624888538606665272082345, −15.74647180765647484941716186144, −15.38416587859560905524047579601, −14.55392582102453488755253123455, −13.57520531729931488776677548936, −12.9347529567250952785163691217, −11.97788755215387363778641091385, −11.37115829984462462986545790959, −10.631592007072763800987741406158, −10.121528844833955758781183322433, −9.50696560052038591901969218558, −8.866281738344197624720857355356, −8.245144532802132731623801605646, −7.56878050747450289243094900639, −6.32069753262087394193708340965, −5.66579679483458016152277937165, −5.115023170829870221772207137641, −4.140102105388281780060784806874, −2.9754528848688676157181729585, −2.51807431446971703184003765161, −1.77879320287419242086270575725, −0.47857441453683779717942658972, 0.89726320516442252271403999300, 1.619653054340747641826786292725, 2.16958362415695329272207737292, 3.00898831169384876872646005910, 4.4031331347538527550989982863, 5.58817118117118694898454516718, 5.80975893959735528410885116238, 6.89578298677761329803938052651, 7.18804267841051527182423239218, 8.005941484602991233972099635632, 8.6951638550864045919499505288, 9.3250568880552966697761458218, 10.33257046730457762938011551786, 10.75689940692643750109401517841, 11.47856489999926778422656033932, 12.18654006859173106859784814511, 13.45240792987260895465532259789, 13.626445487314481915724318465482, 14.25491055604530298842859749858, 15.09689339176832859027343981578, 16.0830249495155196400089995251, 16.66123426393513695923271490714, 17.38959453198594276201115283840, 17.904992503875617821882515243257, 18.230013962670307227537026040454

Graph of the $Z$-function along the critical line